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# Wave mechanics

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### Wave mechanics

1. 1. The Study of Water Waves Breaking December 7th 2006
2. 2. Contents•  Introduction •  Shallow Water Waves •  Deep Water Waves •  Beach Waves•  Breaking Waves•  Solutions •  Analytical •  Experimental •  Computational•  Conclusion
3. 3. Introduction: Water Waves•  Stokes wave train generated by an oscillating plunger•  28 wavelengths away from the generation point, the waves disintegrate
4. 4. The Waves Become Unstable
5. 5. In sight•  In the following slides the depth is 22% of the wavelength.•  White particles are photographed over one period demonstrating the trajectories of the water particles•  A standing wave is created by propagating waves at the left and then having the waves reflected at the far right•  The radius of the circles traced by the particles trajectories decreases exponentially with depth.
6. 6. Shallow Water Waves•  In shallow water waves, the particle trajectories are elliptical versus the circular paths found in deep water waves Deep Water Waves•  Deep water waves are those whose wavelength is less than the height of the body of water
7. 7. Beach Waves•  Biesel noted that, as ocean waves approach the beach the water gets shallower. This causes: •  The amplitude of the waves to increase •  The wave speed decreases •  The wavelength decreases •  The water particles tend to align there elliptical trajectories with the bottom s slope Circles -> Ellipses Floor
8. 8. Breaking Waves•  The changes in the properties of the wave lead to structural instability in a non-linear manner. As the wave approaches the beach the bottom is slowed down while the top part continues forward.•  Thusly, the wave breaks.
9. 9. Solutions•  Analytical•  Experimental•  Computational
10. 10. Analytical•  A parametric solution for a breaking wave has been developed by Longuet and Higgins.•  This solution only describes the flow up to the moment of impact. Another solution involving turbulence is required to describe the aftermath.
11. 11. Parametric representation; branch points•  Consider the flow to be incompressible, irrotational and in two dimensions•  x = ! + i! and z = x +iy are the particle position coordinates•  Assume x and z are analytic functions of complex ! and or t•  x can be expresssed as a function of z if ! is eliminated•  The following suffixed terms represent the partial differentiation with respect to ! or t•  W* is the particle velocity, where W=X!/Z!
12. 12. Say z! = 0 at !=!0, where z=z0 then, near here,z-z0~1/2(!-!0)2z!!!-!0 !(z- z0)1/2, then,x-x0 ~ (!-!0)(z-z0)1/2
13. 13. Boundary ConditionFrom Bernoulli;-2p = (xt-Wzt) + c.c. + WW* - g(z+z*) – 2!,-2 Dp/Dt = (xtt-Wztt) + 2K(x!t-Wz!t) + K2(x!!-Wz!!) + c.c. – g(W+W*)– 2!,whereK = D!/Dt = W*-zt . z!at the boundary condition of the free moving surface,p = 0, Dp/Dt = 0 so,zt = W*, K= 0then,-2 Dp/Dt = (xtt-Wztt) + c.c. – g(W+W*)– 2!,
14. 14. We ve assumed that the flow is Lagrangian and ! is real at the free surface. Now,ztt – g = irz!where r is some function of ! and t which is real on the boundaryit was found that the particle acceleration is,a = D/Dt zt(!*) = ztt(!*) + K*zt!*(!*),K= [zt(!*)-zt(!)]/z! ,Frames of Referenceztt – g = irz!is a non-homogeneous linear diferential equation for z(!,t) with solutions z0(!,t) and z0 + z1(!,t). then,ztt = irz!z1 = z0 -1/2 gt2,
15. 15. The Stokes Corner FlowVelocity potentialX = - 1/12 g2(!-t)3 = 2/3 ig1/2z3/2Longuet and Higgins proved that at the tip ofthe plunging wave there s an interior flow whichis the focus of a rotating hyperbolic flow.
16. 16. i! = !½ t (ztt-g) = z!, Upwelling Flowz = - ½ gt!z! = - ½ gt,W = zt*(-!) = ½ g!,x! = ¼ g2t!,x = - 1/8 g2t!2 = - z2/2tThe free surface is the y-axisVelocity potential! = - x2 – y2 2tthe streamlines are!=- xy = constant, tFor t > 0 the flow represents adecelerated upwelling, in whichthe vertical and horizontalcomponents of flow are given by!x = - x/t, !y = y/tat x = 0 the pressure is constant-py = vt +(uvx+vvy),where (u,v) = (!x,!y) à py = 0-p = !t + ½ (!x2 +!y2) – gx
17. 17. Now, we want the solution to the homogeneous boundary condition. This will describe the flows complementary to the upwelling flow.½ tztt = z!when ! +!* = 0Make z a polynomial, z = bn!n + bn-1!n-1 + … + b0; bn is a function of timebn!n + is of the form At + B and A and B are constant P0 = t, P1 = t! + t2, P2 = t!2 + 2t2! + 2/3 t3, P3 = t!2 + 3t2!2 + 2t3! 1/3 t4. Q0 = 1, Q1 = ! + 2tln|t|, Q2 = !2 + 4t!ln|t| + 4t2(ln|t| - 3/2), Q3 = !3 + 6t!2ln|t| + 12t2!(ln|t| - 3/2) + 4t3(ln|t| - 7/3) z = !n (AnPn + BnQn)
18. 18. Physical MeaningTo understand these flows; consider a linear, cubic, and quadratic flow.
19. 19. Comparison with ObservationOnly the front face of thewave is being described
20. 20. Experimental•  The following is a numerical approach involving coefficients that were experimentally determined.
21. 21. The SetupThe wave s velocityfields were measuredlaser Dopplervelocimeter (LDV) andparticle imagevelocimetry (PIV)
22. 22. Computational•  The computation of a breaking wave acts as a good test to see if the numerical model accurately depicts nature.
23. 23. A Popular ModelIn 1804 Gerstner Developed a wave model whose particle (x,z) coordinates are mapped as so: zx = x0 –Rsin(Kx0-!t)z = z0 +Rcos(Kx0-!t) xWhere,R= R0eKz0 R0 = particle trajectory radius = 1/KK= number of waves != angular speedz0 = !A2/4! A= 2R != 2!/K
24. 24. Biesel improved on this model to account for the particle trajectory stendency toward an elliptical shape.Improving still on Biesel s model was Founier-Reevesx = x0 + Rcos(!)Sxsin(!) + Rsin(!)Szcos(!)z = z0 – Rcos(!)Szcos(!) + Rsin(!)Sxsin(!)Sx = (1-e-kxh)-1,Sz(1 – e-Kzh)sin(!) = sin(!)e-K0h! = - !t + !0x0K(x)!xK(x) = K!/(tanh(K!h))1/2
25. 25. Here:K0 – relates depth to to the angle of the particles elliptical trajectoryKx – is the enlargement factor on the major axis of the ellipseKz – is the reduction factor of the minor axisThese variables range between 0 and 1. They are used to tune the model in order to avoidunreal results that arise from a negatively sloping beach. Waves (-) Beach Floor
26. 26. ConclusionAn accurate model of a wave crashing on the beach could yield beneficial information for coastal structures such as boats, or break-walls.Also,Perhaps a more accurate wave model could assist in the design of surf boards
27. 27. References•  Cramer, M.s. "Water Waves Introduction." Fluidmech.Net. 2004. Cambridge University. 6 Dec. 2006 <http://www.fluidmech.net/tutorials/ ocean/w_waves.htm>.•  Crowe, Clayton T., Donald F. Elger, and John A. Roberson. Engineering Fluid Mechanics. 7th ed. United States: John Wiley & Sons, Inc., 2001. 350.•  Gonzato, Jean-Christophe, and Bertrand Le Saec. "A Phenomenological Model of Coastal Scenes Based on Physical Considerations." Laboratoire Bordelais De Recherche En Informatique.•  Lin, Pengzhi, and Philip L. Liu. "A Numerical Study of Breaking Waves in the Surf Zone." Journal of Fluid Mechanics 359 (1998): 239-264.•  Longuet, and Higgins. "Parametric Solutions for Breaking Waves." Journal of Fluid Mechanics 121 (1982): 403-424.•  Richeson, David. "Water Waves." June 2001. Dickinson College. 6 Dec. 2006 <http://users.dickinson.edu/~richesod/waves/index.html>.•  Van Dyke, Milton. An Album of Fluid Mechanics. 10th ed. Stanford, California: The Parabolic P, 1982.
28. 28. Hydraulic Jump